SOME GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY
PROBLEMS
I. INTRODUCTION
In the solution of railway problems involving the characteristics
of the motive power it is difficult to use analytical methods, principally
because it is impossible to obtain a satisfactory general equation for
the curves of an engine or motor of any specified type. The relation
between speed and tractive effort, for instance, is so involved that
any attempt to obtain a formula leads to assumptions which cannot be
made without seriously affecting the accuracy of the final result.*
This is true not only of the steam locomotive, but also of the various
types of electric motors ordinarily used for train propulsion.
The graphical methods, in contrast with the analytical, form an
accurate and at the same time an easy means of attack applicable
to any possible combination of characteristics and any range of con-
ditions which may be met in practice. It is the purpose of this bulletin
to develop a number of new graphical methods which, in connection
with other well-known ones, aid materially in the solution of such
problems. While most of these were developed in connection with
problems of electric train performance, a number of them are equally
applicable to any type of motive power, a fact which is set forth in
the paragraphs which follow.
The majority of these methods were developed by the writer in
connection with classroom instruction. One of the ways of obtaining
motor performance with varying potential and one for finding the
"effective" value of the motor current are due to Mr. S. Sekine, a
graduate student in Railway Engineering in the University of Illi-
nois, who is also responsible for a portion of the method of plotting
speed-time and distance-time curves.
II. MOTOR PERFORMANCE WITH VARYING POTENTIALI
The performance characteristics of a railway motor are ordinarily
furnished by the manufacturer for the normal potential and are
usually assumed to be accurate under such conditions. Often it is
desirable to find the motor performance when abnormal potential is
impressed on the terminals, since in practice the line pressure is sub-
ject to wide fluctuations, and the motors are always operating at
subnormal potential while the controller is being turned to the full-
speed position.
*See C. O. Mailloux, Discussion on paper by F. W. Carter, Transactions
A. I. E. E., Vol. XXII, p. 165 (1903).
tFor a brief discussion of this topic see Electric Railway Journal, Sept. 18,
1915.
ILLINOIS ENGINEERING EXPERIMENT STATION
The torque produced by a given current in a series motor is
practically independent of the line pressure,* so that recalculation
of this quantity is unnecessary for any ordinary conditions of opera-
tion met with in practice, unless, of course, the field strength is pur-
posely reduced. The only other important variable to be considered
is the motor speed.
In an electric motor the applied pressure is used up in two ways;
a portion overcomes the drop due to the resistance of the windings,
and the remainder opposes the counter e.m.f. generated in the arma-
ture. If the field flux remains constant, the speed will vary in direct
proportion to the counter e.m.f. which is developed. This may be
expressed by the equation
V2_ 2 E-Ir
V1 E1-lr ............ ........ (1)
V, E,-Ir
in which V, and V2 are the speeds when E1 volts and E2 volts are
applied at the terminals, respectively, I is the current flowing through
the armature, and r is the motor resistance, or that portion in the
armature and the circuits in series therewith.
0 50 100 150 200 250 300
MOTOR CURRENT. AMPERES
FIG. 1. VOLT-AMPERE DIAGRAM FOR ELECTRIC MOTOR.
In order to make the calculation graphically it is only necessary
to determine the relative values of E, - Ir and E, - Ir, from which
the ratio of speeds may be found directly. A simple method of
showing the relations between these values is to construct a diagram
with motor volts as ordinates and armature amperes as abscissae, as
*A. M. Buck, The Electric Railway, p. 53.
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
shown in Fig. 1. Since the Ir drop is a direct function of the arma-
ture current, it can be represented for all values of current by the
intercepts on a straight line with the proper slope. This may be
drawn through the origin, but, since we are principally concerned
with the difference between the terminal pressure and the Ir drop,
it is better to draw it from the line of full pressure at the motor
terminals, E,. If the terminal pressure is then changed to E, volts,
it will not affect the slope of the Ir line, but will change its position
so that it begins at the point E,. In each case the counter e.m.f. is
the residue after subtracting the Ir drop, as shown in the diagram.
All that remains is to obtain a graphical relation between V1 and V,,
which is proportional to these values of counter e.m.f. Two methods
of doing this have been developed.
The first method of calculation is shown in Fig. 2. Here the
volt-ampere diagram of Fig. 1 is reproduced, along with the speed-
current curve of the motor, as determined by test or from design
calculations, the axes of current being in the same straight line. The
MOTOR CURRENT, AMPERES
3
O
r
r
D
B
o
MOTOR CURRENT, AMPERES
FIG. 2. CONSTRUCTION FOR OBTAINING MOTOR SPEEDS AT DIFFERENT POTENTIALS.
current scales and their positions along the axis may be chosen as
desired, their relation to each other being immaterial. The speed of
the motor at the terminal pressure E1 is represented by the ordinate
V,. It is desired to find the corresponding value of speed V, at E,
volts and the same current I. Draw a line through A at the value
of current I on the volt-ampere diagram and also through V,. This
ILLINOIS ENGINEERING EXPERIMENT STATION
will intersect the axis of abscissae at some point K. From K draw
the line KB, through the corresponding point B on the volt-ampere
diagram for the same current and the new pressure E2. This locates
V,, the speed at E, volts, at the intersection of KB with the current
ordinate. It must be correct since, by similar triangles,
IV, I'A
Iv i j7 A ....................... (2)
It may be seen from Fig. 1 that I'A and I'B are the values of counter
e.m.f. corresponding to the pressures E1 and E2 at the current I.
It should be noted that a different position of the point K will
be located for each value of current, and in some cases it may be at
too great a distance from the body of the diagram. To obviate this
the relative positions of the speed-current and the volt-ampere dia-
grams may be changed, always keeping their current axes together.
In some cases it is preferable to make the entire construction
on the speed current diagram. The arrangement for this method
is shown in Fig. 3. Here the base of the volt-ampere diagram is
taken the same as that for the speed-current curve, and the propor-
MOTOR E. M. F., VOLTS
60 600
50 500
400
0
S 300
20
§200
10 100
0 0 50
100 150 200
MOTOR CURRENT, AMPERES
FIG. 3. SECOND METHOD FOR OBTAINING MOTOR SPEEDS AT DIFFERENT POTENTIALS.
tional division is made by swinging one set of values of counter
e.m.f. through an angle of 90 degrees, so that EN is equal to OE,.
The two projections of the values of counter e.m.f. will meet at some
point, such as P, and a line drawn connecting P with the origin will
250 3W
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
divide the ordinate and abscissa of any point along it proportionally
to these two values. Then, by projecting the speed at E1 volts onto
this line, the speed at E2 volts and the same current are given by
the corresponding abscissa, and may be carried back through 90
degrees and plotted on the original current ordinate, as shown.
A further inspection of Fig. 3 shows that the locus of the point
P will be a line MN, which passes through N, corresponding to zero
Ir drop, and makes an angle of 45 degrees with the axes. The proof
of this construction is that the Ir drop is the same for a given cur-
rent irrespective of the terminal pressure. For this reason it is
unnecessary to swing mechanically the counter e.m.f. line through
90 degrees to locate P. Draw MN from the intersection N of the
projections of E, and E, (taken at right angles, as explained above).
Any point on the counter e.m.f. line will then give a projection on
MN, as at P, thus saving the preliminary construction.
III. MOTOR PERFORMANCE WITH RESISTANCE
To determine the performance of a motor when a resistance
is inserted in series with the armature, the constructions given in
Figs. 2 and 3 may be used with a slight modification. Fig. 4 is the
1U IDU 200
MOTOR CURRENT, AMPERES
MOTOR CURRENT, AMPERES
2u 30W
FIG. 4. FIRST METHOD FOR OBTAINING MOTOR SPEEDS WITH RESISTANCE.
same as Fig. 2, except that the Ir drop at a different pressure has
been replaced by a line EB representing the drop I(R -- r), in which
R is the external resistance in the circuit. The procedure is the
" 5U
ILLINOIS ENGINEERING EXPERIMENT STATION
same as that explained in the determination of motor performance
with varying potential, and the proof of the construction is identical.
The method of Fig. 3 can equally well be used for determining
motor speeds with resistance, as shown in Fig. 5. Since the IR drop
MOTOR E. M. F., VOLTS
60 600
50 500
4o ,oo
40 400
0
30 2 300
C
20
10 100
0 0 50 100 150 200
MOTOR CURRENT, AMPERES
250 300
FIG. 5. SECOND METHOD FOR DETERMINING MOTOR SPEEDS WITH RESISTANCE.
is not the same, the line MN has a different angle, which is determined
by the relative values of resistance in the two cases; that is, if the
line MN of Fig. 5 makes an angle 0 with the axis of abscissae,
tan = r ......................(3)
R + r"
With this modification the method is precisely the same as that
described above.
IV. STARTING RESISTANCE FOR SERIES MOTORS WITH
RHEOSTATIC CONTROL
In starting direct-current series motors it is usually not sufficient
to reduce the potential at the motor terminals by making different
combinations of motors on the supply circuit. When this can be
done, as may be possible with very small motors, the performance
may be predicted by calculating the performance curves at the lower
potentials, as described previously in this bulletin, or by any other
ordinary method. In general, however, it is necessary to place a cer-
tain external resistance in the circuit, whether or not the potential
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
at the terminals is reduced by any other means. The added resistance
should be just sufficient to give the desired values of starting current
and torque, the one being dependent on the other. As the motor gains
speed, the resistance must be reduced, or the current and the torque
will fall off too much. Of course, unless the resistance can be cut out
in infinitesimal steps, there will be some variation in these quantities,
the range being determined by the allowable difference between the
maximum and minimum values of torque and current.
The simplest method of control consists merely in connecting the
motor or motors to the line with an external resistance in series, the
latter being reduced in steps until finally it is all out of the circuit
and the motors are directly across the line. It is essential to deter-
mine correctly the exact values of resistance to be placed in circuit
on each point of the controller in order that the conditions of current
and torque limits may be met. This can be done quickly and accu-
rately by a graphical method based on those given above.
When the motor is stationary the current which will flow is deter-
mined entirely by the resistances in the circuit, since the effect of
inductance enters only at the instant of connecting to the line, and
there is no counter e.m.f. being developed at the time. Since the
internal resistance of a well-designed machine is quite small, it is
necessary to add a considerable external resistance to keep the initial
current down to a proper amount. The exact value of current desired
depends on the torque needed and on the capacity of the motor
and the connecting wiring. Having determined the required current,
it is a simple matter to find the necessary resistance. This may be
done directly by the application of Ohm's law. Let Im be the maxi-
mum allowable motor current, E the line e.m.f., r the motor resistance,
and R1 the external resistance to be inserted at starting. Then
E
Im - ....... ........ (4)
R, +r
from which R1 may be found at once if the other quantities are known.
As soon as current flows through the motor, a torque is developed, and
the armature will commence to rotate. This will cause the generation
of a counter e.m.f. tending to oppose the e.m.f. of the circuit, so that
the current will be reduced. The torque falls off correspondingly,
and if the action is allowed to continue the performance will be as
shown in Fig. 4 or Fig. 5, the acceleration dropping until the motor
operates at some constant speed. Since it is usually desirable to bring
the motor up to full speed as soon as practicable, it is customary to
reduce the amount of resistance in the circuit so that the accelerating
current will remain near the maximum value. The amount of re-
sistance which should be removed from the circuit at one time is a
function of the total number of steps in which it is to be cut out or
the allowable variation from the mean value of the starting torque.
The latter is the simpler case and will be considered first.
12 ILLINOIS ENGINEERING EXPERIMENT STATION
Assume that the allowable variation from the mean value of
starting torque to give smooth acceleration is 10 per cent. The min-
imum torque will then be approximately 20 per cent less than the
maximum, which latter corresponds to the current at standstill, as
determined by equation (4). As previously explained, the current
will decrease from the instant of starting until it has fallen to the
minimum desired value determined from the proper acceleration.
At this point the counter e.m.f. developed by the armature will have
risen to some value which can be determined readily, since the sum
of the resistance drop, I(R, + r), and the counter e.m.f., Ec, must
equal the line pressure; that is,
E= E + I(R,+ r) ..................(5)
Since the value of resistance has already been found by equation (4),
the value of Ec can be obtained.
When the current has fallen to its minimum value In the resistance
of the circuit should be reduced enough to bring the current up to
the maximum value Im. In order to find this new value of resistance,
it is necessary to determine the counter e.m.f. which will exist after
the change in connections has been made. If the field flux of the
motor remained constant, then, disregarding small variations due to
changes in armature reaction and other causes, the counter e.m.f.
would be the same for any value of armature current. But in the series
motor the field flux is a function of the armature current*, since the
latter also flows through the field. The flux will therefore become
greater when the current is increased by the removal of some of the
resistance. The exact amount of this change depends on the pro-
portions of the magnetic circuits of the motor, and can be determined
from the saturation curve of the machine. For practical purposes
of calculating starting resistance, this method is not available, since
it requires making a special test of the motor. There are, however,
methods which may be used for getting approximate proportional
values of flux which will serve the purpose equally well.
The speed of an electric motor varies directly with the counter
e.m.f. developed and inversely with the field flux. From this it may
be seen that the flux is directly proportional to the counter e.m.f. and
inversely proportional to the speed; that is,
E,
kn ........................ (6)
Jkn
in which 4 is the field flux, n the speed of rotation, and k a constant
depending on the winding, etc.
Since
Ec= E - Ir ...................... (7)
equation (6) may be rewritten
*In case the field of a series motor is shunted, the current through it is
directly proportional to that through the armature, although not equal to it.
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
E- Ir
kn (8)
In the ordinary motor it is not possible to determine with any
accuracy the constant k unless access may be had to the design data.
But, since in the calculation of starting resistances only proportional
values of flux are required, the knowledge of this constant is entirely
unnecessary. Therefore, the following equation may be used with
equal accuracy:
E - Ir
k- ................ ...(9)
If the motor resistance is known, the relation of k( to the armature
current I may be calculated for any current and a curve plotted
if desired.
Another method of getting proportional values of flux depends
on the relation of this quantity to the torque developed by the
motor. In any electric motor the torque is proportional to the
armature current and the field flux; that is,
D - K4 ............... ..... . (10)
where D is the torque of the motor at a current I, and K is a pro-
portionality constant depending on the winding, but not the same
as k in the preceding equation. As before, a curve may be plotted,
giving proportional values of flux for any armature current.
When the current is increased from In to Im by reducing the
resistance in the circuit, the flux increases from (n to 'm. During the
infinitesimal time required for changing the current, it is evident
that the speed cannot change. It must follow, therefore, that the
counter e.m.f. will increase, due to the greater flux. By equation (5),
the new value will be the counter e.m.f., Ecn, at the minimum cur-
rent, I,, multiplied by the ratio of fluxes. The new counter e.m.f.,
Ecm, can then be found as follows:
E c - m = kEco
Ecm e n , ..................(11)
or,
E = ( K ).................(12)
depending on which method was used for getting the proportional
values of flux. For brevity, call this ratio of field fluxes Q; that is,
k4'm Kcm
Q k K .................(13)
k-% KeDn
Ecm= QEc. ..................... (14)
Then,
If the maximum and minimum values of current are to be
reached each time the resistance is changed, then the ratio Q becomes
ILLINOIS ENGINEERING EXPERIMENT STATION
constant for the particular conditions assumed, and the calculation
of resistances is simplified considerably. On the other hand, it may
be advisable to allow different values of current on the various steps
of the controller, in which case the ratio of fluxes must be determined
separately for each point. When the controller is equipped with a
current-limiting device the former condition holds. By the appli-
cation of the above equations the values of resistance for a rheo-
static controller may be calculated.
It is more convenient for the engineer to calculate the resist-
ances by a graphical process, since the use of the equations is some-
what tedious. For this purpose the volt-ampere diagram may be
employed conveniently. In Fig. 6 the volt-ampere diagram of Fig. 1
oo
400
200
00
0
0
MOTOR CURRENT, AMPERES
FIG. 6. DIAGRAM FOR DETERMINING RESISTANCES FOR SERIES MOTOR WITH
RHEOSTATIC CONTROL.
has been repeated, and on it is also plotted the curve of relative
values of flux (k4 or KY) against current. The limits Im and In being
chosen, it is evident that the ratio Q will be constant. If, then, a line
is drawn through the points ým and 4~, cutting the axis of abscissae
at X, the latter will be the intercept of all lines cutting the verticals
through I and I, at points proportional to these values of flux;
that is, in the figure,
, Imn'm ImBm
I -n etc., Q .................(15)
IfA' IA,
since all of the triangles whose apexes pass through the point X
divide parallel lines into proportional parts.
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
Starting with the maximum current Im, the entire external poten-
tial E is used up in overcoming resistances. That is, the line ImE"
represents the IR drop, ImGm being that in the external resistor
and GmE" that in the motor itself. As soon as the armature begins
to rotate a counter e.m.f is developed. When the motor current has
fallen to In this e.m.f. is represented by the ordinate IAn, the line ImE
being drawn through E, for evidently there will be no IR drop with
zero current. It is evident that when the current has reached In resist-
ance must be cut out in one step until the current rises to the max-
imum, Im. Since the counter e.m.f. has a value of InAn when the
current is a minimum, it follows that it must increase by the ratio Q
when the current is increased to Im so rapidly that the motor does
not have time to change its speed. The new counter e.m.f. may be
determined by projecting a line from X through An, intersecting the
line of maximum current at Bm. The counter e.m.f. at this point
is represented by ImBm, the drop in the external resistor by BmGm,
and that in the motor by Gm E". The external resistance to be
employed is found by dividing BmGm by the current Im. The process
may now be continued until all the external resistance has been
removed and the motor is running on the line. This condition is
shown by the line EGm, and from this point on the normal curves
of motor performance apply.
If it is desired to change the current limits at any stage of the
controller operation, the proper resistance can be determined in the
same manner, the location of the point X being varied to correspond
to the proper values of current. For small changes, the location
of X may be assumed constant without introducing an appreciable
error. If a definite number of steps is called for, as by the adoption
of a standard controller, the values of Im and In must be changed
until the exact number of steps is obtained on the diagram. This
must be done by trial, but the adjustment can be made quickly after
a few cases have been solved.
As given above, the diagram has been worked out for a single
series motor. If two motors are to be run in parallel, it is only
necessary to modify the diagram to give the proper values of cur-
rent, remembering that the combined resistance of the machines
is but one-half that of a single motor. For operation with machines
in series the same precautions must be observed, but in this ease
the motor resistance is twice that of a single machine. With these
variations, the diagram can be modified to meet any combinations
of rheostatic control of series motors.
V. SERIES-PARALLEL CONTROL*
In electric railway practice it is customary to operate series
motors in pairs or in groups of motors in pairs. They are ordinarily
*See Electric Railway Journal, Dec. 26, 1914, and Feb. 13, 1915.
ILLINOIS ENGINEERING EXPERIMENT STATION
controlled by the series-parallel method, which involves placing the
two units in series with resistance which is cut out in steps, changing
to parallel with the resistance again inserted, and finally cutting
it out again in steps. Generally the current limits are the same
for both connections, although sometimes they are different in the
series and in the parallel arrangements.
The calculation of the counter e.m.f. and the resistance for
series-parallel control is made in the same manner as for the rheo-
static, except that the precautions mentioned under the former topic
on p. 15 must be observed very carefully. It is usually convenient
to combine the series and the parallel diagrams into one. This is
shown in Fig. 7. The method of construction is the same as for
500
M
2O o
100
0
0 100 200 300
MOTOR CURRENT, AMPERES
FIa. 7. DIAGRAM FOR DETERMINING RESISTANCES FOR SERIES MOTORS
WITH SERIES-PARALLEL CONTROL.
rheostatic control, the difference being that the point S, representing
half potential, is taken as the point for drawing the IR lines while
the motors are in series, and the point E for the same purpose after
the parallel connection is made. It is necessary to interpret cor-
rectly the values of IR drop to determine the resistances. When the
motors are in series the current flowing through the circuit is that
through a single machine, while after they are thrown in parallel
the line current is that for two motors. To determine the series
resistances, therefore, the external IR drop, for instance that on the
first point of the controller, is equal to ImSm per motor, so that this
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
must be doubled to get the total drop in the external circuit. The
correct value of resistance to put in series with the motors on the
first point is then
R -Im ...................(16)
Im
and similarly for any other value of series resistance.
When the connections are changed from series to parallel, the
counter e.m.f. of each motor is InSn just before breaking the circuit,
and ImEm after the reconnection is complete. In series, the counter
e.m.f.'s of the two motors add, while in parallel they do not. The
residue, EmPm, must therefore be consumed in external resistance.
On the first parallel point the resistance must then be
EmPm
R 2Im ............... .. (17)
and so on until the motors are directly on the line. In all other
respects the series-parallel diagram is precisely the same as the
rheostatic diagram previously described.
VI. STARTING RESISTANCE FOR SHUNT MOTORS
The calculation of starting resistances for shunt motors is made
in the same manner as for series machines, the principal difference
oo0
lo
200 E
100
0 100 200 300
ARMATURE CURRENT, AMPERES
FIG. 8. DIAGRAM FOR DETERMINING RESISTANCES FOR SHUNT MOTOR
WITH RHEOSTATIC CONTROL.
being that, since the field is supplied by a circuit in parallel with
the armature, the field flux is practically constant at a given poten-
tial for all values of armature current. It is, therefore, unnecessary
to determine any change of flux when the resistance is reduced.
ILLINOIS ENGINEERING EXPERIMENT STATION
The diagram for calculating graphically the values of armature
resistance is given in Fig. 8 for a single motor. This diagram is
somewhat similar to Fig. 6, except that the lines representing the
change from one point to the next are not drawn through a single
point X, but are all parallel to the base. The method of getting the
resistances from measurements on the diagram is the same as pre-
viously described. For series-parallel control a similar scheme may
be followed. It is not illustrated here on account of the infrequency
of the use of series-parallel control with shunt motors.
VII. PLOTTING SPEED-TIME CURVES
A number of methods have been proposed from time to time
to reduce the labor incident to the plotting of speed-time curves for
railway trains. The analytical solutions all depend on producing
equations representing the characteristic curves of the motive power;
and, on account of the difficulty of determining separately the equa-
tion of the curve for each separate motor or locomotive, general
solutions giving the average of a large number of machines have
been used. Although this is satisfactory for approximate calcula-
tions in which extreme accuracy is not required, as in preliminary
estimates, it is not suitable for problems involving a particular
machine. For such cases graphical or semi-graphical methods are
usually resorted to if a solution more rapid and less laborious than
that obtained by the point-by-point construction is desired.
Of the graphical methods, the first one which was satisfactory
was that developed by Mr. C. O. Mailloux.* The construction there
described is of a high degree of accuracy, and is so simple that it
may be readily applied. It has the disadvantage of requiring a
number of charts on which the graphical solution is based, and which
take considerable time for preparation. Although the method saves
labor when a large number of determinations must be made for the
same equipment, the time taken for construction of the charts is a
serious disadvantage when but a few runs are to be calculated. A
scheme intended to obviate the latter difficulty was devised by Pro-
fessor E. C. Woodruff,t in which the separate charts are replaced
by diagrams drawn directly on the motor curve-sheet. Although the
work of plotting is somewhat less than in the Mailloux method, and
the intermediate calculations are all on the single motor curve-sheet,
considerable time is still required for plotting the diagrams needed
in the determination.
From time to time constructions have been developed for accom-
plishing portions of the desired result, and these may be considered
useful for modifications of the original methods just described. They
*Notes on the Plotting of Speed-Time Curves, Transactions A. I. E. E., Vol.
XIX, p. 901 (1902).
fGraphic Method for Speed-Time and Distance-Time Curves, Transactions
A. I. E. E., Vol. XXXIII, p. 1673 (1914).
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
simplify and in some cases reduce the labor incident to the graphical
calculation.
The plan herein proposed is a graphical solution which possesses
the accuracy of the original ones, while at the same time it eliminates
nearly all of the intermediate steps. The calculations are all based
on fundamentally correct principles, and the results may be deter-
mined as closely as desired within the limits of accuracy of the ordi-
nary methods of plotting.
The acceleration produced by a known tractive effort is given in
the following equation:
F
A =91.1(1-r)T. . ......... (18)
91.1(1 + r)T
in which A is the acceleration in miles per hour per second, F the
net tractive effort of the motor in pounds at the wheel treads, T the
weight of the train in tons per motor, 91.1 the force needed for unit
acceleration of translation alone, and r the ratio of force required
for the acceleration of rotating parts to that for translation. When
extreme accuracy is not necessary, equation (18) can be replaced
by the simpler statement
A 10(19)
A= ......... ......... (19)
100T
in which the rotating parts are assumed to take approximately one-
tenth the force necessary for acceleration of translation. It is evi-
dent from these equations that for a given weight of train per motor
the acceleration produced is directly proportional to the net tractive
effort.
The force available for acceleration, or net tractive effort, is
the residue of the total torque of the motors, after reducing to the
speed at the wheel treads, subtracting the force for overcoming train
resistance and curve resistance, and subtracting or adding the force
for going up or down grades. The size and type of the cars making
up the train being known, and the profile given, it is a comparatively
simple matter to determine these quantities. Train resistance may
be calculated from tests or by any one of a number of well-known
formulae, as, for example, that developed by Mr. A. H. Armstrong:
R 50 .0 0.002 aV2 n -1
R= /W 0.03 V - W 1 -J. . .... (20)
V W W 10
in which R is the train resistance in pounds per ton, W, the weight
of the train in tons, V, the train speed in miles per hour, a, the
projected cross-section of the train, and n, the number of cars in
the train. This is probably as accurate as any general equation devel-
oped for passenger cars. For freight trains, other equations should
be used.*
*See Bulletin 43, Engineering Experiment Station, University of Illinois.
"Freight Train Resistance," by Edward C. Schmidt.
ILLINOIS ENGINEERING EXPERIMENT STATION
Grades require an additional tractive effort of 20 pounds per
ton for each per cent of up grade, and correspondingly less for down
grade. Curve resistance is quite difficult to determine, but may be
assumed from 0.5 pound to 2.0 pounds per ton per degree of curva-
ture. After making the proper subtractions and additions to the
gross tractive effort given by the motive power, the force available
for producing acceleration, or net tractive effort, is obtained.
0 . 100 150
CURRENT, AMPERES
200 250
FIG. 9. RAILWAY MOTOR CHARACTERISTIC CURVES.
Since the mathematical operations for getting the net tractive
effort are addition and subtraction, the calculation may be made
by a graphical process. This has been explained by many previous
writers, so that it is not necessary to repeat it here.* It is worth
noting, however, that if the train resistance is subtracted directly
on the diagram, the residue represents at once the net tractive effort
for level track, while if plotted separately the process of subtraction
is rendered more difficult, requiring the use of a scale or a pair
*C. O. Mailloux, Notes on the Plotting of Speed-Time Curves, Transactions
A. I. E. E., Vol. XIX, p. 901 (1902).
E. C. Woodruff, Graphic Method for Speed-Time and Distance-Time Curves,
Transactions A. I. E. E., Vol. XXXIII, p. 1673 (1914).
A. M. Buck, The Electric Railway, p. 39.
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
of dividers, in addition to the coordinate scales of the chart. Grade
and curve resistances being single-value functions (i.e., not changing
with speed), they may be represented by horizontal lines on the
chart, either increasing or decreasing the ordinates of tractive effort.
The manufacturer's performance curves for a certain electric
railway motor are given in Fig. 9, with the addition of the train
resistance and grade and curve resistances for use with a particular
train or car. The net tractive effort curve gives directly the acceler-
ating force for level track; and for other conditions the base may
be moved up or down as required. It is to be noted that the values
of resistance are plotted in terms of force per motor, so that if, for
example, the equipment consists of four motors, the values on the
chart will be one-fourth of the total.
The net tractive effort having been determined, the acceleration
produced may be found from equations (18) or (19). These equa-
tions show that if the tractive effort is plotted as an ordinate and
the quantity 100T from equation (19) as an abscissa, the slope of
the line connecting the origin with the point thus determined is a
measure of the acceleration to the same scale. The actual value of
the slope is not important; it depends on the units chosen for the
coordinates of the speed-time curve.
In plotting the speed-time curve, the most satisfactory way is
to take an increment of speed, AV and, knowing the value of accel-
eration, A, to determine the corresponding increment of time, At. It
is this method which has been elaborated by all writers and which
is the basis of the present article. Since
AV
A ........................(21)
then
1
At= AV . ......................(22)
This equation is the basis of the former methods of graphical deter-
mination of speed-time functions. In Mailloux' method, a chart of
inverse values of A and of integral multiples of these values is plotted.
An inspection of equation (22) shows that if an increment AV equal
to unity is taken, At is the reciprocal of A, so that it may be taken
directly from the chart. A somewhat similar method is followed
by Woodruff, who, however, combines the reciprocal curve and the
chart of accelerations on one sheet.
A comparison of equations (18) or (19) and (21) shows them
to be of precisely the same form, so that they may be equated as
follows:
AV F
. . . . 00T ......... ......... ....(23)
using the simpler form of the expression given in equation (19).
using the simpler form of the expression given in equation (19).
ILLINOIS ENGINEERING EXPERIMENT STATION
Equation (23) makes it evident that if the time is taken to the same
scale as 100T, then the speed must be to the same scale as F, the net
tractive effort. If a particular scale for time values has been decided
on, the scale for T, which is immaterial to the construction since
only one point need be found, is determined as in the following
paragraph.
Let Vo be the scale of ordinates; namely, the number of miles per
hour per unit of ordinates, and to the corresponding scale of abscisse,
as required for the speed-time curve. From equation (21) the slope
of the acceleration line produced with these unit values may be
determined. The scale for 100T being arbitrary, then if it is chosen
so that the slope of the line is the same as that for unit acceleration.
it makes possible the direct construction of the speed-time curve. The
diagram, Fig. 10, shows the arrangement. With ordinate OB and
FIG. 10. CONSTRUCTION FOR PLOTTING SPEED-TIME CURVE FROM
TRACTIVE EFFORT-SPEED CURVE.
abscissa OA on the speed-time curve, each equal to the unit selected,
the corresponding acceleration is
OB (miles per hour)
OB (m e --- = A (miles per hour per second) ... (24)
OA (seconds)
The discussion shows that a definite amount of force, Fo, is
required to produce this acceleration in a given weight of train.
Selecting any suitable scale of tractive efforts, as NN', this force
may be represented by the ordinate NP. It is evident that if a
straight line is drawn through P parallel to the acceleration line OM,
cutting the horizontal axis at Q, the length NQ will represent the
quantity 100T to the proper scale. This is proved by equation (23)
and the similarity of the triangles OMA and QPN. The same
equation shows that any other value of tractive effort, as NR, will
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
produce an acceleration represented by the slope of the line QR, the
corresponding location of the speed-time curve being OL, drawn
with the same slope. The values of tractive effort for which the
acceleration is determined may be read directly from the tractive
effort curve, plotted against speed, as in Fig. 10, or from the curve
of tractive effort plotted against current, as in Fig. 11.
The construction of the speed-time curve is now evident. If the
train is started with any constant current, I, (Fig. 11), the accelera-
Y
FIG. 11. CONSTRUCTION FOR PLOTTING SPEED-TIME CURVE FROM MOTOR
SPEED-CURRENT AND TRACTIVE EFFORT-CURRENT CURVES.
tion produced is represented by the slope of QR,. The line OV, is
then drawn with the same slope. The constant current can be main-
tained up to the speed S,, so that the line OV, should be continued
until it reaches this ordinate at the point V1. This line and point
are on the speed-time curve to the desired scale. With further
increase in speed, the tractive effort will decrease, as indicated by the
curve, and the acceleration will be correspondingly less.
Consider an increment of speed, 'V= V2,-V1. This cor-
responds to a decrease of tractive effort from T, to T,. If the incre-
ment is taken small enough that the variation in force is practically
along a straight line, the average tractive effort, acting continuously
for a time At, will produce an increase in velocity AV. If, then, the
tractive effort at the mean speed,
V + /2AV = 1/2 (V1 + V2) .............. (25)
is taken and projected on OY, at R2, the slope of the line joining
this point with Q is the average acceleration during the increment.
A line drawn through V, parallel to QR, will pass through the
point V2 at the end of the increment AV. The location may be made
conveniently by projecting S2 parallel to the axis of abscissae, and
noting the intersection V,. If the increment has been taken suffi-
ciently small, this is a point on the curve, and not a tangent; for the
ILLINOIS ENGINEERING EXPERIMENT STATION
tangent to the curve at the mean ordinate would not pass through
the points V, and V,, but would be parallel to the line drawn through
them. The magnitude of the error due to this assumption is fully
discussed by Mailloux.* It is shown that the error is so small as
to be negligible in ordinary calculations if AV is not too great.
The construction outlined in the last paragraph may now be
continued for the remainder of the acceleration period until the train
reaches constant speed. A smooth curve drawn through the points
located in this manner is the true speed-time curve; and the accuracy
may be made as great as desired by proper choice of the speed
increments.
For the coasting portion of the curve, the train resistance may
be plotted to any horizontal scale, the ordinates being the same as
those for motor tractive effort. In fact, the ordinates representing
train resistance, which are plotted down from the gross tractive effort
curve, may be stepped off with dividers and transferred to the line
OY to determine the corresponding retardation. Speed increments
may be taken as before, and the coasting curve plotted. For the
braking curve an ordinate corresponding to the braking force must
be obtained and added to the train resistance. In this manner the
entire speed-time curve may be determined.
VIII. PLOTTING DISTANCE-TIME CURVESf
In constructing distance-time curves, a number of methods may
be used. Mailloux determines distance by means of the device known
as the "integraph," which is a convenient and accurate way. If
such an instrument is not available, a planimeter may be used, making
partial integrations over portions of the run, so that enough points
may be located to draw the curve. This is a much slower process,
although of practically the same accuracy as the former. In the
absence of any other device, the area of the curve may be determined
by making the plot on coordinate paper and counting the small
squares included by the diagram. Woodruff uses a series of curves
representing distance covered at average speeds, which may be used
in estimating the distance passed over during the various increments.
A method which is at least as accurate as any of the purely
graphical constructions mentioned is described in the following
paragraph:
Assume any convenient scale of distance to be used for plotting
the distance-time curve on the same sheet as the speed-time curve.
Referring to Fig. 12, let OB represent unit distance, say one mile.
This same ordinate corresponds to a speed of V miles per hour on
*Transactions A. I. E. E., Vol. XIX, p. 988 (1902).
tThe process described for plotting distance-time curves is a general
method of graphical integration, and may be used for the construction of integral
curves for any function whatever that may be represented by Cartesian graphs.
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
the speed-time curve. If the train continues in motion at a velocity
of V miles per hour for hours, the distance covered will evidently
be one mile. Since the speed in such motion is constant, the rate of
C-
TIME
FIG. 12. CONSTRUCTION FOR PLOTTING DISTANCE-TIME CURVE FROM
SPFED-TIME CURVE.
covering distance, or the slope of the distance-time curve represent-
ing the run, is a straight line. Lay off a length OC on the time axis
equal to - hours, and erect the perpendicular CDK at C. A diag-
onal line connecting O and D will then measure the distance traversed
when the speed is represented by the ordinate CD = OB. In other
words, OD is the correct distance-time curve for a constant speed
OB = V. For any other time, the distance covered will be pro-
portional, and will be represented equally well by the ordinate of
the line OD up to that time. Since distance is proportional to the
product of speed and time, the distance covered at any other velocity
1
during the time hours will be represented by an ordinate equal
to that speed.
This construction may be utilized in plotting the distance-time
curve from the speed-time curve as follows. Take the average velocity
during any time increment and project the ordinate representing it
on the line CDK. The intercept on the line joining the projection
of this average speed with the origin included within the limits of
the time increment measures the distance covered. For instance, the
first portion of the speed-time curve, terminating in the point V1, has
been made at a constant acceleration. The average speed during the
E
ILLINOIS ENGINEERING EXPERIMENT STATION
increment is 1/2(V1+O). Locate the point E on the line CDK so
that CE = 1/2V1. Connect 0 and E by the straight line OE. The
time increment at the point V1 intersects this line at F. This is a
point on the distance-time curve since, for uniform acceleration, the
distance s is
Vo + V(
s- At ...................(26)
2
which is a fundamental relation. For the next increment, from
V, to V,, the average velocity, 1/2 (V, + V2), is represented by the
ordinate CG, and the line OG determines the slope of the distance
curve during this period. A line FH, beginning at the point F and
drawn parallel to OG will, therefore, determine the point H on the
distance curve at the end of the time increment. This construction
may be continued until the entire distance-time curve is located. A
smooth graph passing through the points thus plotted is the true
distance-time curve. As in the case of the speed-time curve, the
points located are actually on the curve and not on tangents. The
construction is accurate so long as the deviation of the speed-time
curve from a straight line is negligible during each increment under
consideration.
It is not claimed that this method of determining distance is
more accurate than the use of the integraph or the planimeter, but
that it is of more ready application, and gives results which are as
accurate as are ordinarily obtainable within the limitations of curve
plotting. The error can be made as small as desired by taking incre-
ments of time of such magnitude that the speed-time curve is prac-
tically straight during any one of them, as explained.
IX. APPLICATIONS OF GRAPHICAL METHOD FOR SPEED-TIME
AND DISTANCE-TIME CURVES
A problem frequently met with in railway service is the determi-
nation of the exact points of cutting off power and of applying
brakes in order to make a run of fixed distance in a given time. The
solution may be made by the application of the speed-time and dis-
tance-time curves. To do this the braking portion of the speed-time
curve may be plotted backward from the end of the run and the
corresponding distance curve located, while the distance curve for
acceleration is plotted forward from the zero point. A period of
coasting must be interposed which will satisfy the operating require-
ments; namely, one which will allow braking to be included at the
normal rate and also reach the desired point for the end of the run.
In order to show the method, a complete speed-time and dis-
tance-time curve will be drawn. While the entire plot is given for
straight level track, the modifications for various combinations of
grade and curve may be made as suggested in the foregoing para-
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
graph. A diagram drawn by this method is shown in Fig. 13. The
run comprises an acceleration with the motors, followed by a period
of coasting, and lastly by a period of braking. This is the simplest
form of run ordinarily used, although it is possible to eliminate the
coasting, applying the brakes immediately after cutting off the power.
0
5.
Q
bj
a
vl
I
cl
TIME, SECONDS
FIG. 13. COMPLETE SPEED-TIME AND DISTANCE-TIME CURVES.
The latter practice, however, is not usual. The various parts of the
run are determined independently and afterward connected together
as indicated in the following paragraphs. The plot of Fig. 13 checks
with that produced by an analytical determination within the limits
of accuracy of the cross-section paper used; and the graphical con-
struction has the further advantage of requiring only a set of triangles
or a parallel ruler when the same scale of ordinates is used for the
speed-time curve as that given on the motor characteristic curve.*
The braking rate is usually assumed constant. A speed-time
curve for this portion of the run may be plotted backward from the
end, as in Fig. 14, and the corresponding distance-time curve deter-
mined.
The coasting speed-time curve is independent of the acceleration
and the braking, for during this period the train is acted on solely
by the force of train resistance and the incidental resistances present
due to the track conditions. For a given profile the coasting speed-
time curve may be determined from the weight of the equipment and
the train resistance equation. It may be drawn graphically by the
methods of Fig. 10 or Fig. 11, the motor tractive effort being replaced
by the train resistance per motor (i.e., the total resistance per train
*It is often undesirable to plot the speed-time curve to the same speed scale
as that of the motor performance. In such a case the time corresponding to a
certain Increment of speed may be found directly by laying off a right triangle, the
hypotenuse of which is parallel to the acceleration line. Since this triangle may be
plotted to any scale whatever, the accuracy may be as great as desired. From the
successive speed and time increments thus found, a speed-time curve may be plotted.
The distance-time curve may be laid out in a similar manner.
ILLINOIS ENGINEERING EXPERIMENT STATION
divided by the number of motors). It should be remarked that a
resistance is a negative force, and should, therefore, be plotted down-
ward from the base. The acceleration produced will be negative
unless the force due to a down grade is such as to equal or exceed
TIME. SECONDS
FIG. 14. METHOD OF DETERMINING PROPER POINT FOR CUTTING OFF POWER.
the negative force of train resistance. A separate speed-time curve
for coasting may be plotted on tracing paper or other transparent
medium and the corresponding distance-time curve located, as shown
in Fig. 15.
TIME SECONDS
FIG. 15. COASTING SPEED-TIME AND DISTANCE-TIME CURVES.
Since the distance-time curve is the first integral of the speed-
time curve, an abrupt change in the slope of the latter corresponds
to a point of inflection in the former, or merely to a change in its
curvature. The coasting distance-time curve must, therefore, be
tangent to both the accelerating and the braking portions of the
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
distance-time curve for the run. For further proof of this it may
be noted that, since the slope of the distance-time curve is a measure
of the speed, this slope must be the same for either curve at the
point where the two portions of the speed-time curve join. This fact
makes possible the following method of accurately locating the points
of cut-off of the current and application of the brakes.
The tracing of the coasting distance curve (Fig. 15) should be
laid over the curve of distance while accelerating (Fig. 14) with the
axes of coordinate parallel, so that the two curves are tangent at
some point. The tracing should then be slid along, keeping the axes
parallel, until the coasting curve also becomes tangent to the braking
distance curve. The points of tangency thus determined correspond
to the cut-off of the current and the application of the brakes. These
points having been determined and the distance-time curve during
coasting transferred to the plot of Fig. 14, the tracing of the coast-
ing curves may be moved parallel to the axis of ordinates until the
two axes of abscissa coincide. The coasting line may now be traced
on Fig. 14, locating definitely the remainder of the speed-time curve
and producing the complete diagram of Fig. 13.
In practice, it is usually convenient to have a number of coasting
curves, corresponding to different conditions of grade and track
curvature, to cover all the variations liable to occur. Such a series,
plotted on a sheet of tracing cloth or transparent celluloid, forms
a templet for the location of the principal points on the speed-time
and distance-time curves for runs of definite length, making the
graphical construction of much greater value in preliminary calcu-
lations to determine the size of motors required for a given service.
The graphical method of plotting speed-time and distance-time
curves described is equally good for use with any kind of motive
power. All that is necessary is to get the relation between speed
and tractive effort connected by a graph which can then be used
for determining accelerations in the manner outlined. The applica-
tion is so obvious that it need not be further elaborated.
X. HEATING VALUE OF A VARIABLE CURRENT.
The rating of all electrical apparatus depends to a considerable
degree on the heating of the active parts. This is especially true
in the case of railway motors. One of the principal sources of heating
is the resistance of the conductors. The heat produced in a wire
carrying a current is proportional to the square of the current multi-
plied by the time during which it is acting. In general, the value
of current in a conductor is not fixed for any considerable period,
but is constantly changing. If the variation follows some known
law, the effect of the current in producing heat can be found by a
comparatively simple mathematical analysis; but if the current is
changing in some casual or variable way, the evaluation is not easy.
ILLINOIS ENGINEERING EXPERIMENT STATION
The latter condition holds true in the case of the electric railway
motor cycle. Here the current is a maximum at the instant of start-
ing, after which it gradually falls to a minimum, and is then cut
off entirely while the train coasts and comes to rest. The variation
is further complicated on account of the occurrence of grades, curves,
points where the speed must be reduced, and other special conditions
of operation.
Railway motors are usually rated by the current which can be
carried continuously, or for a stated period, with a temperature rise
above the surrounding air considered safe.* To determine whether
or not a motor is large enough for a given service, the variable
current must be evaluated to find whether it is above or below a
safe amount. The method usually employed is to plot the curve of
current taken by the motor against time and from this construct
another curve of values of current squared. The integral of the latter
curve, divided by the total time of operation, is the square of that
current which, applied continuously for the same time, will produce
the same loss in the conductors.
As ordinarily applied this method is cumbersome. It requires the
use of a table of squares or some similar method of calculation, so that
the new curve can be plotted from the original current values. To
obviate the necessity of squaring a large number of values, another
plan has been devised, which requires the replotting of the current
curve in polar coordinates. The effective current can be obtained
by this method without the need of squaring the ordinates of the
current curve.
The entire argument in favor of the use of the polar diagram
for finding the effective motor current is that it is less laborious than
to plot the curve of squared values of current. Two methods, both
of them entirely graphical, will now be described for plotting the
latter curve, which is more easily prepared by these methods than
the polar diagram. The other operations involved in the determina-
tion of the effective current are essentially the same for either this
or the polar method. The curve of squares of current plotted on a
rectangular base has the further advantage that it can be put on
the same sheet with the original current curve, thus rendering unneces-
sary the use of a separate chart and making possible an easier coordi-
nation of the values than when the diagrams employed are so different
in character as the rectangular and the polar graphs.
In Fig. 16 consider a scale of natural numbers, ON. Corre-
sponding to these it is desired to construct another scale of natural
numbers such that a certain ordinate O'M is equivalent to ON. It
is evident that the square of ON may be represented by the area
*For further information regarding the methods in vogue for rating railway
motors, see Standardization Rules of the A. I. E. E., 1915 edition.
tFor a proof, see A. M. Buck, The Electric Railway, p. 136.
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
ONBA enclosed by the rectangle having each of its sides equal to
ON. Since the scale of squares is chosen so that O'M is numerically
equal to the square of ON, it may equally well be stated that it rep-
resents the area ONBA. The problem is to find the ordinate along
FIG. 16. METHOD FOR SQUARING NUMBERS GRAPHICALLY.
O'M corresponding to the square of some other value, as OD on the
original scale. It has been seen that the scale O'M may be consid-
ered to measure areas, so that the discussion resolves itself into finding
the ordinate along O'M which will represent the area ODEH, which
is the square constructed on the side OD. If a rectangle with the
base OA can be found with an equivalent area, its ordinate will be
the value sought.
Referring to Fig. 16, construct the diagonal OB of the large
square, and continue DE to meet AB at the point C. Connect C
with O, cutting HE at F. The geometrical construction gives
ON==AB==OA=NB ................ (27)
OD=HE=OH=DE=AC ......... (28)
OG=HF=AK .....................(29)
HE AC OD
AB AB ON(3
HF AC OD (
HE AR N............... ....(31)
HE AB ON
Hence
OD AC
OG= HE X = OD X...................... (32)
ON AB
ILLINOIS ENGINEERING EXPERIMENT STATION
Therefore
Area OGKA = Area ODEH ................ (33)
Since
Area OGKA = OG X OA
=(OD X6_ )X ON
= OD ..................... ..... ............... (34)
Hence O'G', the numerical equivalent of OG, represents OD to the
scale of O'M.
The application of the method is obvious. It is only necessary
to construct a square at any point on the current-time chart, with a
side such that some value of current and its square are represented
by the same length of side. Any value of current, corresponding
to OD, should be projected on the diagonal OB and also on the
side AB of the square. When the projection AC on AB has the
point C connected with 0, the line CO will intersect HE in the
point F. This is the ordinate for the curve of current squared, and
may be carried back to the proper position above or below the cor-
responding value of current. With a small amount of practice the
calculation can be made with great rapidity, for the construction
lines can be very largely omitted, only the intersections being required
to find the proper values.
FIG. 17. PARABOLIC CURVE FOR SQUARING NUMBERS.
An alternative method to that just described is to plot a curve
between the natural numbers and their squares, the latter values
being represented by convenient ordinates. An inspection of Fig. 16
indicates that the locus of the points F is a parabola whose principal
axis is ON and which passes through the point B. In practice it is
found simpler to make the diagram of the opposite form, as shown
in Fig. 17. The parabola OFB is of the form
x= ky2 ............ ......... . (35)
GRAPHICAL SOLUTIONS OF ELECTRIC RAILWAY PROBLEMS
Consider a line OEB drawn through the origin. Ordinates cut off
by this line, as HE, are proportional to the abscissae, as OH. Corre-
responding ordinates on the parabola are proportional to the square
roots of the abscissae. Therefore HE is equal to HF on such a scale
that AB is represented by the same ordinate, AB. The construction
holds for any line OB intersecting the parabola.
In order to apply the method just described, the parabola OFB
and the straight line OEB should be plotted on some transparent
medium, such as celluloid. The templet thus made may be slid along
the curve of current with the axis OA coinciding with the base line
of the current curve, until the parabola intersects the current curve
at the proper point; the square of that ordinate will then be found
directly under or over the value of current. This can be repeated
an indefinite number of times until sufficient points are obtained to
plot the curve of current squared. From this the effective current
may be obtained, as explained above.
Since the plotting of points as obtained by the parabolic curve
may be difficult when the base lines OA coincide, since holes will
have to be pricked through the templet, the method may be modified
to permit the construction being placed on an ordinary celluloid
triangle by moving the axis of the line OEB upward through a
suitable distance. This is shown in Fig. 18. Here the base line
FIG. 18. APPLICATION OF PARABOLIC CURVE TO A TRIANGLE.
for the parabola is OA, the edge of the triangle; while that for the
diagonal line has been transferred to O'A', at a distance 00' above the
other axis. All the ordinates along O'E'B' are therefore displaced
by the amount O'B'. This will not occasion any difficulty in the
subsequent calculations, since the value obtained for the area of the
current squared curve will be too great by an amount equal to
00' multiplied by the length of the diagram. As the area is to
be divided by the base to find the mean ordinate, the calculation can
be made without reference to the constant, and the value of 00' sub-
tracted from the mean ordinate for the current squared curve.